Explanation:
(1)
sin(3θ) = sin(2θ)
Double and triple angle formulas:
3 sin θ − 4 sin³ θ = 2 sin θ cos θ
sin θ ≠ 0, so divide both sides by sin θ:
3 − 4 sin² θ = 2 cos θ
Pythagorean identity:
3 − 4 (1 − cos² θ) = 2 cos θ
3 − 4 + 4 cos² θ = 2 cos θ
4 cos² θ − 2 cos θ − 1 = 0
Quadratic formula:
cos θ = [ 2 ± √((-2)² − 4(4)(-1)) ] / 2(4)
cos θ = (2 ± √20) / 8
cos θ = (2 ± 2√5) / 8
cos θ = (1 ± √5) / 4
θ = 36°, so it's in the first quadrant. Therefore, cos θ > 0.
cos θ = (1 + √5) / 4
(2)
sin³(2x) cos(6x) + cos³(2x) sin(6x)
Multiply and divide by 4:
¼ (4 sin³(2x) cos(6x) + 4 cos³(2x) sin(6x))
Power reduction formula:
¼ ((3 sin(2x) − sin(6x)) cos(6x) + (3 cos(2x) + cos(6x)) sin(6x))
¼ (3 sin(2x) cos(6x) − sin(6x) cos(6x) + 3 cos(2x) sin(6x) + cos(6x) sin(6x))
¼ (3 sin(2x) cos(6x) + 3 cos(2x) sin(6x))
¾ (sin(2x) cos(6x) + cos(2x) sin(6x))
Angle sum formula:
¾ sin(8x)